What is Modular Congruence? | Congruence Modulo n, Modular Congruence Definitions

What is modular congruence? What is congruence modulo n? We go over two equivalent definitions of this relation in today’s math lesson! This is your first step to learning modular arithmetic, and can serve as a great gateway to learning about equivalence classes as well! We also briefly discuss modular congruence with negative numbers in the video lesson.

We say that a is congruent to b mod n ( written a ≡ b (mod n) ) if and only if a and b have the same remainders when divided by n. So for example, when we divide 12 by 5, we have a remainder of 2. When we divide 27 by 5, we also have a remainder of 2. Thus, 12 is congruent to 27 mod 5. We can observe the remainders by noticing 12 = 5*2 + 2 and 27 = 5*5 + 2. See? They both have remainder 2!

An equivalent definition says that a is congruent to b mod n if and only if n divides a - b, which means that a - b is an integer multiple of n. For example, 20 - 2 = 18, and 6 divides 18. Thus, 20 ≡ 2 (mod 6).

Also, it is important to note that modular congruence is an equivalence relation! Just like “equals”! This means that modular congruence is reflexive, symmetric, and transitive!



I hope you find this video helpful, and be sure to ask any questions down in the comments!

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